I struggle with the notation of the following problem or at least I am unsure what is asked for here:
Consider the Basis $B = \{ \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}, \begin{pmatrix} -2 \\ 1 \\ \end{pmatrix}\} $ of $R^{2}$ and fine a linear mapping $D: R^{2} \to R^{2}$ by $Df_{1}=-f_{1}; Df_{2}=3*f_{2}$.
Give the matrix representation $A_{BB}$ and $A_{EE}$ of $D$, where $E$ is the standard basis of $R^{2}$.
So first of all I do not understand what $A_{BB}$ does really stand for? Why $BB$? I started out with computing the change of basis matrix $C$ for $R^{2}_{B} \to R^{2}_{E}$ and its inverse.
So $$C = \begin{pmatrix} 1/5 & 2/5 \\ -2/5 & 1/5 \\ \end{pmatrix}$$ and $$C^{-1} = \begin{pmatrix} 1 & -2 \\ 2 & 1 \\ \end{pmatrix}$$ and the matrix $A_{B}$ that represents the mapping $D$ w.r.t. $B$
$$A = \begin{pmatrix} 11/5 & 8/3 \\ 8/5 & 4/3 \\ \end{pmatrix}$$
But I do not know how to proceed since I am unsure what $A_{BB}$ stands for. Thanks!